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ANTENNA LABORATORY 
Technical Report No. 49 

EVALUATION OF CROSS-CORRELATION METHODS 
IN THE UTILIZATION OF ANTENNA SYSTEMS 

by 
Robert H, MacPhie 

25 January 1961 

Contract AF33(616)-6079 
Project No. 9-(13-6278) Task 40572 



Sponsored by: 
WRIGHT AIR DEVELOPMENT CENTER 



Electrical Engineering Research Laboratory 

Engineering Experiment Station 

University of Illinois 



Urbana, Illinois 



ENGINEERING LIBKARY 

ACKNOWLEDGMENT 

The author is very grateful for the many helpful suggestions from 
Professor Deschamps. The discussions with Professor Lo^ Mr. Craig Allen 
and other members of the Antenna Laboratory were also of considerable' help, 






ABSTRACT 

The theory of the cross-correlation of signals of two or more antennas to 
produce antenna product patterns is reviewed and presented in a generalized 
form. The problem of distortion of amplitude modulation by the multiplicative 
processes is considered and a system to remove this distortion by filtering 
is proposed. An analysis is made of the correlation system output when in the 
presence of a primary desired signal there is a secondary interfering signal 
incident on the antenna system from a different direction. It is shown that 
the rejection of this interference does not depend entirely on the product 
pattern itself. Indeed^ if the interfering signals amplitude is relatively 
low, the interference in the system output is determined mainly by a low-level 
interference pattern which is essentially the sum of the factor patterns of 
the individual antennas rather than their product. Some ostensibly good 
product patterns have associated with them very poor low-level interference 
patterns. The analysis considers the generalization to more than one inter- 
fering signal and more than one pattern multiplication as well as the effect 
of time averaging (filtering) which is sometimes possible, for example, in 
the case of Radio Astronomy. 

Use is made of the negative sidelobes of the product pattern to suppress 
the integrated effect of average background noise. It is shown that for any 
arbitrary noise background with the signal known to be absent the time average 
of the total contribution of this noise in the system output can be reduced to 
zero. Then when the signal is transmitted its time averaged effect alone will 
be observed in the system output. It is assumed that the noise from any given 
direction is a quasi-stationary random process which is statistically independent 



of the noise from any other direction. 

A discussion of the noise limitation on the formation of arbitrary 
patterns from a two element array by means of multiplicative operations is 
presented and it is shown that any background noise will generally cause a 
shift in the apparent direction of arrival of the signal as 'seen" by the 
two element product pattern. The formation of an arbitrary pattern from 
this basic two element pattern with its error due to the shifting action 
of the noise does nothing to reduce this error. Rather^ it only serves to 
emphasize the apparent direction of arrival of the signal and if the noise 
is relatively large the error between the apparent direction and the true 
direction can be considerable. 



CONTENTS 

Page 

1, I rr reduction 1 

2, The Multiplication of Antenna Patterns bv Signal Cross-Correlation 3 

2„1 Two Antenna Case 3 

2„2 Demodulation and Multiplicative Distortion 5 
2„3 N Antenna Case and Removal of Amplitude Modulation Distcr- 

^ion by filtering 6 

3, Analysis of the Effect of More Than One Plane Wave Incident en 

the Antenna System 13 

3 1 0;+put of An Additive System with Two Plane Waves Incident 13 
3„2 Output of a Two Antenna Multiplicative System with Two 

: nndpp' Plane Waves 14 
3 3 Output of a Two Antenna Multiplicative System with R 

Interfering Signals 22 

3.4 '•.■*" erf erence in the Case of More Than One Multiplication 24 

4c Suppr-? -" <~>n ft T he Effect of Average Background Noise by Use 

of the Negative Sidelobes of the Product Pattern 28 

5„ The NToise Lim ition on the Formation of Arbitrary Patterns 
from a Two Element Interferometer by Means of Multiplicative 
Operations 36 

6, Cone 1 us i on' 42 

Reference? 44 



ILLUSTRATIONS 



Figure 
Number 



Page 
1. Mul tipli ca T j en of Two Antenna Patterns 4 



2. Frequency Spectra of Signals at Successive Stages of the 
Process of Removing Undesirable Modulation on the Signals 

to be Cross-Correlated. 8 

3. System for Removing Modulation from All but the Last 
Product Signal for the Case of Multiplication of N 

Antenna Patterns 10 

4. Comparison of Product Pattern and Low-Level Interference 

Pattern for the Compound Interferometer 19 

5. System for the Suppression of the Average Value of the Output 

of a Compound Interferometer Due to an Arbitrary Distribution of 
Background Noise 34 

6„ Phaser Representation of the Signal Plus Noise Output from a 

Two Element Multiplicative Array. 40 

7. <a) Interferometer Pattern of Signal Alone, (b) Interferometer 
Pattern of Noise Alone, (c) Interferometer Pattern of Signal 
Plus Noise^ (d) Synthesized Pattern of Signal Plus Noise 42 






INTRODUCTION 



For some time cross-correlation techniques have been used to obtain receiving 
antenna patterns which are effectively the product of the patterns of two or 

more different antennas. Practical examples of such systems are the Mill's 

1 2 

Cross and the Compound Interferometer both used in Radio Astronomy. In the 

former, two long, mutually perpendicular antennas of lengths L and L in the 
form of a cross, have their signals cross-correlated to produce a power pattern, 
with a pencil beam, roughly equal to the voltage pattern of an antenna whose 
area is L L , The Compound Interferometer places a uniformly weighted aperture 
beside a simple interferometer and the cross-correlated output combines the 
sharpness of the interferometer pattern with the directivity of the aperture. 
If both are of length $ (hence the system is of length 2i ) , the product 
power pattern is equal to the voltage pattern of a uniform aperture of length 
4i. 

In this report the product pattern for the general case of two antennas 
separated by a distance i and with complex patterns is presented Then the 
further generalization to the multiplication of N antenna patterns is made, 
and a system to accomplish this is proposed. In this connection the problem of 
demodulation is considered. The multiplicative operations cause distortion of 
the modulating signal and it is shown that this multiplicative distortion can 
be eliminated (in theory) by filtering. 

The problem of rejecting interfering signals from arbitrary directions 
other than that of the desired signal is more complicated than in the case of 
additive arrays. It will be shown that the product pattern alone does not 
describe fully the rejection of interference. Another pattern, which in general 



depends on the relative directions of the primary and interfering signals, has 
to be introduced. In the case of a single cross-correlation the pattern is 
essentially the sum of the factor patterns of the two antennas rather than 
their product It will be shown that some good product patterns have associated 
with them rather poor low-level interference patterns „ More generally, if the 
signals of N antennas are cross-correlated, in the system output there are N-l 
distinct interference patterns all of which must be analysed when interference 
rejection is required. 

The product pattern, unlike the power pattern of additive antenna systems, 
has negative sidelobes and in Section 4 it is shown that, they can be used to 
eliminate the effect of background noise in the time-averaged output of the 
correlation system. The proposed scheme will work only when it is known a 
priori when the signal is present and when it is absent,, In addition the 
noise from any direction must be quasi-stationary and statistically inde- 
pendent of noise coming from any other direction„ 

Finally an analysis of the noise limitation on the formation of arbitrary 
patterns from a two element array by means of multiplicative operations is 
presented. It is shown that errors introduced by the noise into the two element 
product pattern cannot be reduced in the formation from i t of patterns of 
ostensibly much greater resolution. 



2. THE MULTIPLICATION OF ANTENNA PATTERNS 
BY SIGNAL CROSS -CORRELATION 



2,1 Two Antenna Case 



A diagram of the multiplication system for two antennas is shown in Figure 
1. The signal from antenna A is 



A(9,t) = Re A(0)e Jwt volts (1) 



and that from antenna B is 

B(6,t) = Re B(e)e" jf3isin6 e Jwt volts (2) 

= Re B (Q) e JCOt volts (2a) 

where 9 = angle which the propagation vector of an incident plane wave makes 
with the normal to the line joining the two antennas, 

t = time in seconds, 

u> = angular frequency of the incident plane wave in radians per second, 

i = spacing between the antennas in meters, 

P = propagation constant in radians per meter, 

Re indicates 'the real part of". 
The pattern coefficients A(Q) and B (0) are in general complex. Note that the 
effect of the spacing i between the antennas is the phase factor e " K 
which is absorbed into B.(0)„ 

Now as is shown in Figure 1, the signal from antenna A enters a frequency 
shifter (a nonlinear device) together with a local signal at frequency oj . 
The output is filtered so as to pass only the difference frequency (oj = to - co ) 



LOCAL 

PUMP 

SIGNAL 



(J 



ANTENNA A 



9 




ANTENNA B 



¥ 



W 



i 



> x 



A(0,<U) 



B^(0,W) 



FREQUENCY 
SHIFTER 

AND 
FILTER 



A(0,o/) 



It 



MIXER 



W. 



i 



Pp<0,W |f <) 



SYNCHRONOUS 
DETECTOR 



I 



R,(0,f) 



'' '• Multiplication of Two Antenna Patt 



erna 






signal which is simply the signal from A shifted in frequency to co . Then this 
signal and that from B enter a mixer which is a second nonlinear device 
whose output contains among other components a signal at frequency to . By means 
of a synchronous detector the amplitude of this signal can be obtained and it 
is given by 



p (e, i) = Re A(e) B*(e) o> 

ft 

where indicates the complex conjugate quantity. In trigonometric form, Equa- 
tion (3) can be written as 



p (e, i) = :A(e) B(e>l cos[pisine + a (e> - p(e)l (3a) 



where a(G) = Arg A(Q) and (3(9) = Arg B(Q). Since P (Q,4) is the product of two 
voltage patterns it varies as the power of the incoming waves and consequently 
any comparison with additive antenna systems patterns will be with the power 
patterns of those systems. We note incidentally from Equation (3) that if 
/ = and A(Q) = B(Q), then the product pattern degenerates to an additive 
systems power pattern given by 

P o (0) = |A(e)^ 2 (4) 

I 

2.2 Demodulation and Multiplicative Distortion 

I — . 

If the incoming plane wave is modulated, with amplitude modulation for 
example, then the output of the system will be 



P 2 (0, F(t),l) =[14 mF(t)] 2 Re A(e) B*(e) (5) 



= [1 + 2mF(t) + m F(t) ] Re A(e) B*(0) (5a) 



where m = percent modulation, (0 < m < 1), 
F(t) = modulating signal. 

2 2 
Hence there is square law distortion (m F(t) ) which can be significant for 

large m. This problem also occurs in additive systems which employ square law 

detection. Gray* has shown for the simple case of sinusoidal modulation that 

for less than 10$ square law distortion m must be less than 40$ which is not 

too severe a restriction. However, if N antenna patterns are multiplied together 

(as will be shown in the next section) the output is proportional to 



P N (G, F(t)) = [1 + m F(t)] N P (0) (6) 



As N increases the restriction rapidly becomes intolerable; for N = 6 and 10$ 

2 2 
distortion from only the m F(t) term in the binomial expansion, m must be 

kept less than 8$, In the following section a system for multiplying N antenna 

patterns which removes this multiplicative distortion is proposed. 

2,3 N Antenna Case and Removal of Amplitude Modulation Distortion by Filtering 

If there were N antennas the spectrum of the signal at the terminals of 
each would be that of the carrier and its modulation while the amplitude and 
phase of the signal would be determined by the antenna's pattern and its loca- 
tion. Now since any one of the N antennas signals is modulated in the same way 
as any other, nothing would be lost if the modulation were removed from the 
carrier ol - • 1 1 but one of these signals. Not only will there be no loss of 
information bill L1 Will be shown that this actually eliminates the multiplicative 
distortion brouL'M aboul by the repeated cross-correlation of the antenna 

'•8 . 



1 Applied Electronics, Wiley, pp. 748-740 



Let the N antennas have terminal voltages given by A (0, t,F(t)), 

1 

A (0,t,F(t)) A (9,t,F(t)) where 

k N 



A, (G,t,F(t)) = [1 + mF(t)] Re A (0) e "J Cpi k sin0-ct) 
£ k k 



= [1 + mF(t)] Re A (0) e'^ 1 (7) 

k 



For simplicity we consider the case of colinear antennas whose patterns are 

functions of alone. The more general case of non-colinear antennas with 

"two-dimensional" patterns (Q,§) is of no greater difficulty as far as the 

multiplicative operations are concerned In Figure 2(a) are shown the spectra 

of A (0, t, F( t)) and of the local signal at frequency co . The separation of 

1 
the local signals frequency co from the carrier's at to is A co and is chosen 

soas to be slightly larger than the spectral width of one of the carrier's 

sidebands. Hence, the output of the first frequency shifter (FS) will contain 

the difference frequency at A co along with the modulation spectrum of the signal 

This difference frequency can be held constant by an automatic frequency control 

device (AFC). The low frequency output spectrum is shown in Figure 2(b). A 

high Q filter tuned to frequency A co will remove the unwanted modulation on the 

new "carrier" leaving a single frequency signal whose amplitude varies as the 

antenna pattern A. (0), (see Figure 2(c)). The frequency shifting operation 

1 
to a much lower frequency is made necessary by the difficulty of removing the 

modulation from an RF carrier by direct filtration. 

A second frequency shifter (FS) ? this time with a local signal at frequency 

U = co - 2 A co is used to return the unmodulated "carrier" at A co to the RF 



SIGNALS INTO FIRST FREQUENCY SHIFTER 

RADIO 



(a) 



FIRST PUMP 

FREQUENCY . ... n/ ,. Q . 
FREQUENCIES 4 ♦ A^.FM.e) 



nrv . 

J. cj ^^ 



(b) 



.1 LOW FREQ. OUTPUT FROM FIRST FREQ. SH 

♦ A{(A£J,F(6)),e) = A{«J,F(a»,G) SHIFTER DOWN 
y<"~"^ /^N IN FREQUENCY TO AW 



FTER 




|A^(AW,e) /MODULATION SPECTRUM F(U)\ 
\REMOVED BY HIGH Q FILTER/ 



A6J 



(d) 



i SIGNALS INTO SECOND FREQ. SHIFTER 
| A^(A6),9) 

1 



At) 



2ND PUMP 
FREQUENCY 



(e) 



T 2 2 Ad) 



SIGNALS OUT OF 2ND FREQ. SHIFTER 

A {| (6J-3A^J,e) 



SUPPRESSED 
CARRIER 



A.(6J-A6J,0) 
■li 



ii SIGNALS INTO FIRST MIXER 



(f) 



CJ-3AO) Gi^ GO-ACJ O) 



t A < 



(g) 



W-3A6) 
OW MIXER 

2 



a., (6),F(6)),e 



CJ-ACJ 0) 



P 2 AlA^I' |P 2 

i/ir\17ir\ 



A<0 2AOJ 3A6J 



(h) 



"P (ACJ.9) /MODULATION SPECTRUM F«J) 



V REMOVED BY HIGH Q FILTER; 



Figure 2. 



FREQUENCY, RAD./SEC. 
PROCESS CAN BE CONTINUED BY STARTING AT STAGE (d) ABOVE AND 
USING A. {CJJ{Q),Q) IN THE NEXT MIXER OPERATION 

BCy Spectra of Signals at Successive Stages of the Process of 
Bmovlng f.'ndosi rable Modulation on the Signals to be Cross-Correlated, 



range. The FS output will be a new carrier at frequency oo with sidebands at 

oo - A oo and oo + A co. At this point it is necessary to suppress the carrier 

(there is no technical difficulty in doing this), Then the two sidebands whose 

amplitudes are proportional to A (Q) are mixed with the modulated signal from 

1 
antenna two, A (0, t,F(t)) as is shown in Figure 2(f). The low frequency 

2 
difference output spectra are shown in Figure 2(g). Again an AFC device is 

used to hold this output constant in frequency. A filtertuned to A co passes 

the product signal given by 



P 2 (9,t, i 1 ,i 2 ) = Re A i (0) A i *(0) e 



j Aw t 



(8) 



and which has all its modulation removed. 

The process can be continued by returning this signal to RF as was done 
in stages 2(d) and 2(e) (see Figure 2). Then by mixing with A ,(0, t,F(t)) and 
filtering one obtains 



P 3 (0,t,i 1 ,i 2 ,i 3 ) = Re A i (0) A i (0) A jf (0) e 

-I *-i O 



j A oo t 



(9) 



which again has all the modulation removed. A diagram of the system is shown 
in Figure 3. The process is continued until the (N - 2)ND unmodulated pro- 
duct is formed, i.e., 

N-2 

j A w t 



P N _ 1 (0,t,i i ,...i N _ 1 ) = Re 



A A (0) A *(0) 
4 x 2q-l 2q 



A (0) e^ 

N-l 



if N is even and > 3, and 



(10) 



Vl^^l'-W =R 



N-l 



n a ( Q ) a *(0) 

4 2q-l 2q 



j A oo t 



if N is odd and > 2 r (in the above formula i is zero), Then, instead of 




c 
bo 

•H 
CO 



O 

3 
T3 
O 

0, 



W 



3 

J2 



3 


•H 


■o 


r ( 





a 


a 


•H 




■^ 


bfl 


H 


e 


3 


•H 


3 


> 







<H 


e 


o 


0) 




ad 







a 


h 


Ed 





U 


>H 






0) 


a 


d 


u 


+J 


M 




■ 


Fi 


>) 





03 


<M 



3 
ho 

•H 



11 



shifting this signal back up to RF, we mix it directly with the modulated signal 

from the last antenna A (0, t,F(t)), The modulated RF difference frequency is 

N 
if the mixer output signal at this frequency is fed into a synchronous detec- 
tor along with a reference signal also at co taken from the local source, then 
the detector output will be 



N 
2 



^(0, F(t) £ 2? l 3 ,...l }i ) = [1 + m F(t)] Re U A < 0) A 

2q-l 



(0) A, *(Q) 

2q 

(11) 



if N is even and > 1 and 



P N (0, F(t), i 2 ,i 3 , ...,1 N ) = [1 + m F(t)] Re 



f N-l 



q 2i A i (e) A % <°> 

4 X 2q-1 *2q 



A iN (8) 



if N is odd and > 2. 

The output has no multiplicative distortion and consequently m need not be 

limited to say, less than 40$ as with a square-law detector. The practical 

success of the system depends on the filtering efficiency which in turn depends 

on the modulation spectrum. The noise problem should be alleviated by the 

filtering but the extra multiplicative operations will clearly degrade the signal 

to noise ratio. Finally, mention should be made of amplification which probably 

will be necessary if N is large. The most convenient place for an amplifier 

'would be immediately after the filter that removes the modulation from the 

pseudo-carrier at A co. A narrow band low noise parametric amplifier could be 

ised. 

It should be noted that the pattern P ( 0, F . __ . , i_, . . , ,i ) is proportional 

N (t) 2 ' N 

;o the Nth power of the electric field strength of the incoming plane wave, 
tence the equivalent "field strength pattern" (voltage) is 



T> t Q V t +■ \ 



\ — ID (d T?f + \ 



e » 



1 
,N 



l 1 CM 



12 

and the equivalent power pattern is 

2 

p N p (e,F(t)i 2 , ....i N ) = P N ( Q ,F(t),i 2 , ... i N ) N (is) 

Any comparison with conventional additive patterns should be made on this 
basis. In the rest of this report only the power pattern will be used and the 
superscript will be omitted. 



13 



3. ANALYSIS OF THE EFFECT OF MORE THAN ONE PLANE 
WAVE INCIDENT ON THE ANTENNA SYSTEM 



In the preceding analysis, only a single plane wave incident at the arbitrary 
angle 9 was considered. In this section it will be shown that when two plane 
waves are incident at angles 9 and Q , the output of the synchronous detector 
of the product system is in important respects different from the output of the 
conventional square law detector of additive systems. We let one of the plane 
waves of unit amplitude at an angle of incidence 9 be the desired signal. The 
other, of complex amplitude Se \ and angle Q , is either due to a discrete, 
remote, noise source or a secondary component of the desired signal which 
arises when there is more than one transmission path from the source to the 
receiver (multipath propagation). 
3.1 Output of An Additive System wit h Two Plane Waves Incident 

In an additive system the received terminal voltage before detection is a 
linear superposition of the two signals 

C(t;l,9;Se , %) = C(t;l,9) + C(t;Se ,S ) 



= Re]c(9) + Se'^1 CCQ^ \ e jwt (14) 



The usual detector > a square law device, has a low frequency output proportional 
to 



. t , 2 2 2 .t 

Ca^iSeW^ = !c(9)^ + S C(9j + 2S Re C(0; C(Q) e" j5i (1! 
l ■ ' i i 1 



)r alternatively 



3 

Welsby and Tucker have obtained some similar results in which they compare 

multiplicative systems with additive systems employing linear rather than 
square law detection. 



14 

I i ^l I 2 I I 2 2 1 I 2 , 

|cu,e;Se J , e 1 )| = |c(e)| + s |c(e 1 )| + 2s|c(e) c(e 1 )|cos l 1 -Ke)+4(e 1 ) 

(15a) 
where £{Q) = ARG C(9) . 

o 

If the pattern is normalized with the primary signal incident at = i 

the beam angle of the pattern, Equation (13a) can be written as 



c(se J , 1 e 1 '> 



= 1 + s 



0(9^ 



+ 2s|c(e i )|cos(l 1 -t(0°)+^(e i )^ (16) 



C(G 1 ) 

where c(9 ) = , The last two terms of Equation (16) represent interference 

and its rejection is seen to depend on c(0 ) alone. 

3.2 Output of a Two Antenna Multiplicative System with Two Incident Plane Waves 

The output of the synchronous detector in the two antenna multiplicative 
case and with two incident plane waves is naturally more involved than that of 
the additive system. It is of the form 



P 2 a,9;Se j ^ 1 9 1 ;i) = ReJA(e)B i *(e) + S 2 Af e^B^^ 



+ s(A(9)B i *(9 1 ) e ' l + A(0 l )B i *(9) e * 



(17) 



or alternatively (and with some rearranging) 



,h 



P (l,e;SB , 9,;i) = A(6) B(Q) cos r : p/sinn-:a(0.'-(3<0> 

& 1 



s a(Oj > B(e 1 )|cos lysine +a(e 1 )-p^e 1 )] 



A(9) B(9)|cofl p;sinG-ta(0)-|i(9)] ■ 



A(Q i | 






a(e 1 )-p(e) 



l I n6 a(6 I -p(6) 



B(9j ) 



bo) 



cos| pislnSj -i a(0)-p(e,)] 



i i no n(f)J |ii n ' 



(17a) 



15 



Now as in the additive case, if the pattern is normalized with the pri- 
mary signal incident at 6 = 0° Equation (15a) becomes 



P 9 (Se J , 1 Q t) = 1 + s 2 ia(9.)b(e i ) 



cos[ pisine Td(e )-p(e )] 
~" c M[a(0° > ^(M o 5 TT " ~~ 



+ S ', ia(6, ) 



cos(e i ^a(e 1 )-p(o°)) 



II cos(a(0°)-p(0°)) 



+ ib(e,) 

i J- 



cos(pisin0 1 -^ 1 ^a(O°)-p(e i )) 



cos(a(0O)-(3(0°)) 



(18) 



J 



A(e 1 ) 



B(e 1 ) 



where 0.(0^ = ^^J . b 'V = j^ • 

Comparing Equations (16) and (18) we note that the basic difference is in the 

last terms which is the major interference term when S <* 1. In the additive 

case (Equation (16)) the rejection of interference depends on the voltage 

pattern c(0 ): the worst case occurs when £ = C(O)-C(0.,) and the interference 
1 11 

term is 2S c(0 )| , In the multiplicative case (Equation (18.)) the low-level 
interference is proportional, not to the product pattern itself but to the sum 

of the normalized factor patterns | a(0 )| and lb(0 )| each of which is multi- 

1 i 1 

plied by a cosine term. Consequently the rejection of this first order ( - s) 
interference by a multiplicative system cannot be determined by the product 
pattern at all. It will be convenient to define the pattern associated with S 
as the low-level interference pattern since it predominates when S is small and 
& is negligible Hence we let 



,ar0 )| cos(£ -q(0 1 )-(3(O°)) * IbO,)! cos(p.fsin0 -S.'crtcft-pfe)) 



cos(a(0°) - (3(0°)) 



This expression is clearly equal to or less than 



ja(0 i )i -» jb(0 1 ) 



(19) 

but the 



2cos[ a(0°)-p(0°)] 
exact nature of the pattern is quite complicated for this general case. Now 



16 



for any given system, i is fixed and the pattern is a function of the two 
variables G-, and £ „ Likewise for any given Q there will be an £ which 



ma 



ximizes the magnitude of L(Q ,§ ,i). To determine this we differentiate with 



respect to £ and equate the derivative to zero. 



8 L<Q i ,£ v O 1 



-laCe^l sin[^ 1 +a(e i )-p(0°)]+|b(e i )| sin(pisine i -S 1 +a(0 o )-p(6 1 )) 



cos(a(0°)-p(0°)) 



= 



(20) 



from which we obtain the relation 



la(0 )lsin(a(0 )-p(O°))-ib(0 )lsin (pfsin G.+a(O°)-p(0 )) 

i- e i ,i; |" ra(e 1 )!cos(a(e 1 )-p(o°)T+ib(e 1 )lcos(piiin e 1 +a(o°>-p(e 1 )) 



-p<e r 

(21) 



Now there are two values of £,(Q,i) (in the range [ 0, 27T]) that satisfy the above 
equation, namely § (6,i) and £ (0 ,i) + 77. Substituting these values of £ 
into the expression for L(0 , 1, -O gives the worst possible patterns L(0 £ ,i ) 
and L(0 |.+ 7T, i), If the patterns a(0 ) and b(0 ) are real, Equation (21) 
reduces to 



€ j (0 I) ■ tan 



b(9 ) sin (P i sin ) 
a(0 ) + b(0 ) cos (P 1 sin Gj) 



(21a) 



and the expr< lor the worst patterns are 



'.'V V " 



a(9 1 )+b(9 1 ) cos(Pisin Oj) ! 2 + | b(Q ] ) si n (Pi sin ) . ' 



a^Oj) + b(0 ) cosfPisin 9, ) 



(22) 



cos (1 (9j , i )) 






17 



ue^ l I+ ir,i) 



a(8,)tb(eJ cos(Pisin ) I 2 + J b(0 n ) sin(Pi sin n K 2 

1 _1_ 1_ L _ 1 1 J 

a(0 ) + b(0 ) cSsTPisin ) 



(22a) 



cosf£ (0 I) + TT) 



But cos (I (0 , I ) + 77) = -cos (| (0 i)) and hence 



L(e x , Ij + ir,i) = -L(0 r e p I) 



(23) 



Subsequently we will consider only 



L o (0 r i) = L(0 r e r i) 



(24) 



knowing that the alternative solution L(0 £, + 77 ? f ) is simply its negative. 
If i = there is a further reduction to 



L (0 ) = - [a(0 ) + b(0 )] 
o 1 2 1 1 



(22b) 



Equation (22) can be put into a quite symmetric form by letting 



M(0 



, I) = a(0 ) + b(0 ) cos(P I sin ) 



(25) 



and 



N(0 r i) = b(0 ) sin(P I sin ) 



(26) 



Then Equation (22) becomes 



W U -2 



M(e x , i) 2 + n(0 i) 2 



cos 



tan 



.1 N( 6p i> 
M(0 £) 



(27) 



18 

We can therefore conclude that some apparently good product patterns 
might have quite unacceptable low level interference patterns. As an example 
of this we consider the Compound Interferometer mentioned in the introduction. 
It consists of a uniformly weighted aperture of length i, adjacent to a simple 
interferometer also of length i. A diagram of the system is shown in Figure 4, 
The normalized factor patterns are 



sin 



P | sin 9 \ 



a(e 1 ) 



. _v 2 

P i sin 0, 



b(9 1 ) 



= cos 



"'P i sin 9. 



Hence the product pattern is proportional to 



P I sin 0. 



sm 



p 2 ( 8i , D---PT 



P i sin A 



sin 9 n 



cos 



V 



cos (P I sin ) 



sin 



/'P £ sin fl 



p 2 ,e 1 - i "' -tn 



P i sin 9, 



sin 9, 



where x = — - — 



sin(4x) 

4x ' 



(28) 



But the worsl l ->w level interference pattern is 



L (x) 

o 2 



x 2 , , „ v 2 

cos x cos 2x f (cos x sin 2x) 



sin x 



+ cos x cos 2x 






, s i n x 
. / • cos x cos 2x 

tan ' - 

c:ds x sin 2x 



(28a) 



19 



O) 

C/5 




V 



( 



\ 






r 

\ 






V 



CT 



< 

I- 
O 

Q 
O 

cr 

Q. 



UJ 
O 

z. 

Ul 

cr 
ui 
Li- 
ce: 

UJ 



UJ 
> Z 

uj or 



5 

o 




. t= 



4 



) 



0) 
as 

a, 

0) 

u 

c 
a> 

i« 
a> 

0) 



CD 

> 

M 

o +j 



•c 



a 



a) 0) 

C w 
CD -u 



•o 

c 

3 

o 
o, 
s 
o 
o 

CD 



o o 
en <h 

•r-l 

h 

IS 

s 

O 

O 



0) 

3 
CUD 

•H 



20 
The two patterns are plotted in Figure 4 as functions of x. It should be 

noted that this graph shows only the normalized patterns (each with its max- 
imum equal to unity). In any particular case, the relative magnitudes of the 

two patterns depend on the value of S. For example, suppose that S = 0.25 

5 
and corresponds to x = — 77. The output signal to interference ratio for 
1 8 

the worst case would be 



R(0 1 S) = — 

S P 2 (9 1 , 1) + 2S L o (0 p I) 



(.25) (.127) + 2(.25) .395 



s = .25 
~ 577/8 



00794 + .1974 



(29) 



= 4.88 



It can be seen that the low-level interference pattern dominates in this case 

because of its high sidelobes which are only 3 db down at x = +77, + 277 ( +377 etc. 

2 

Note that in some cases S P (0 , SL ) < and hence the worst case of 

interference would occur when 2S L (0,, 1) were negative also. If L (0,, i) 

o 1' o 1 ' 

is not negative as in the above example, we would select the alternate worst 

pattern -L (0, , 1 ) = L (0 , , £ _ + 77 t ) (see Equation (23)). The worst possible 
o 1 o 1' I 

signal to interference ratio for the system, without regard to sign, is therefore 



R <G., i, S 

f> 1 ' ' 



)l = 



S I P.O., 1)1 + 2S|L (0 i ) 
2 1' o 1 ' 



(30) 



By way of comparison, the worst case for an additive system would be 



R (6 , S)| ■ 

8 2 |c(9 1 )| 2 + 2810(8] » 



(31) 



21 
In the latter equation it is seen that high directivity of c(0 ) also means 
a large signal to interference ratio while the high "directivity'' of P (0 , I ) 
in the former equation can be negated by L (0 S.) to give a low net signal 
to interference ratio. 

It is important to realize^ however, that this situation obtains only 
when the interfering signal is at its worst possible phase with respect to 
the desired signal. Usually the interference phase £ is a time varying quantity 
with the probability of any one value of £ (t) in the interval [0, 27T] being the 
same as any other. Hence, the average of the low-level interference when taken 
over a suitably long period of time approaches zero, i.e.. 



1 im 
T -r 



A(0 X ) 
Me)" 



cos (P I sin + |(t) + a(Q ) - (3(e)) 
~cos (f3 £ "sin + cl(6) - (3(e) 



(32) 



P«9 , ) . cos ((3 i sin 0, - £(t) + a(0) - (3(0 )) 



1 

ft©; 



1 r___ _ 

"cos Wl sin + o-(0) _r T(0)) 



dt = 



In practice this means that a low pass filter would be used to reduce the 
low-level interference signal in the system output. This filtering process 
is quite feasible in Radio Astronomy (hence the usefulness of Compound Inter- 
ferometer and indeed of the Mill's Cross), since only the steady state flux 
of energy from each point on the celestial sphere is required. For instan- 
taneous reception of modulated signals or for any application where a lengthy 
time averaging is not practicable, the low-level interference pattern must 
be considered. 



22 

3.3 Output of a Two Antenna Multiplicative System with R Interfering Signals 



If in addition to the desired signal of unit amplitude incident at the 

th 
angle 0, there are R secondary interfering signals^ the q of which has 

amplitude S e and direction , the output of the square law detector 

q q' 

of the additive system will be 



j£. 



j&, 



6(1, S ie , i; ...; S R e , R ) 



R # -jb 

= |'C(6)| + 2Re fo'e) 2 S C (0 )e q 

q-1 q q 



r r : 

. Z SsSc(0)c(0)e 

-, n q J" q r 

q=l r=l 



6 -I ) 



(33) 



The analogous output from the cross-correlation system is 



P 2 (l, 0, S x e e, . 



ji f * R R 

• • s o e R , D ^) = Re i A (e) B , (e) + s s s s 

R ' R £ q r 

q=l r=l 



£(e > - l(e„) 

A(9 ) B . (0 ) e q 

q f r 



R # -jl(9 ) 

r + A(0) 2 S B. (0 ) e 
r i r 
r=l 



t-j€ e i 



+ B. (0) Z S A(0 ) e 

^ , q q 

q=l 



(34) 



A comparison oJ » h<;se equations with those for a single interfering signal 
shows thai th< ten again has the low-level interference pattern 

van by Equation N9). The rejection of Interference Li 



23 



S . <1 for all q will depend primarily on this pattern which, as was shown 

q 

in Section 3.2, can be quite poor even when the product pattern itself is 
good. The step from R discrete interfering signals to a continuous back- 
ground of interference is quite straightforward, and the rejection of low- 
level interference in this case also depends on the low-level interference 
pattern. 

A major hindrance in the design of a satisfactory interference pattern 
is the interferometer factor cos (P i sin 0) due to the separation i of the 
two antennas Its oscillations between +1 and -1 cause high sidelobes, 
especially if bi0 ) has high sidelobes also. 

Indeed, it is quite easy to show that it is impossible to get a product 
pattern and a low-level interference pattern that both possess low sidelobes 
if one of the factor patterns does not have them. 

If I is quite large the cosine factor in the noise term 

cos[P I sin + a(e ) - P(9 )] 

s lace^ b( 0l )' Ho^To") -(3(o o )i 



from Equation (18) will oscillate rapidly between -1 and +1 as varies 
Hence, for a number of values of this noise term will be given by 

!a(0 ) b(0 )1 

S _ i y. sHa(0 ) b(0 )| 

cos[a(0°) - P(0°)] L X 



and if b(0 ) has high sidelobes for these values of the noise term will be 



large. It can be reduced only by making ia(0 )| small, much smaller th 



an 



Ibce^ 



However, the low-level interference noise term is given by 



24 

cos (I + a(0 ) _ (3(0°)) cos (Pi sin - | * a(0°) - (3(0 )) 

s ; a(0 ) __ _i 1 + |b(0 )| L_-i_ _ .J. 

1 cos[a(0°) -P(0°)] cos;a(0 ) -(3(0°)] 

As £ varies this noise component varies also. It can be shown that the max- 
imum value is at least 

S \ ibO.M - la(0 )| r 

___^_ i i__i_ S - [b(0)| - |a(0 n )| ' 

cos[a(0°) - (3(0°)] [ 1 l 

To minimize this component we therefore must make ia(0 )\ srf (b(0 , ) ■ , 
and this is not what is required for the suppression of the other noise compon- 
ent. It follows that one or the other of the noise components can be minimized 
but not both. 

3.4 Interference in the Case of More Tnan One Multiplication 

The case of the multiplication of N antenna patterns with R interfering 
signals is obviously an involved situation. For N even the product pattern is 



N/2 . jPl sin R j| 

V 1 °< S l *, °! - *> - " A 2q -l (9)e * = V \q-l (9 j ) 

q=l v J=l 



(35) 



e jPi 2q-l Sin 9 j 



-jPi .me r -jft -^ 2q .i Bin e k > 

A 2q (6) e ^IS k e A 2q (0 k ) e 



Bj way Oi b N i 4, R ■ 1, and let the patterns be real. Even with 

the expression that results when put into trigonometric 



25 



P (1, 0: Se , . i i i ) = A (0)A(0)A(9)A (6)cos<Pl sine-Pi sinG+Pl sine) 



1 * "7, 



cos iPi sine -Pi sin0+Pi' sin0) 
2 3 4 



A (0 ) 
2 1 



/A (0 ) 

S ! -r- 7 -r— cos(Pi sin0-Pi sin© 
A ( e ) 2 3 

1 1 



+ Pi 4 sin0 + l 1 ) + a~7q7- cos(Pi 2 sin0 1 - Pi 3 sin0 4- 0| sine - ^) 



A (0 ) 

t A / Q x ~ cos (Pi sin0 - Pi sin0 1 + Pi sine 4 ^) 



W \ 

r-7r T - cos (Pi sine - Pi sine + Pi sine 1 - (L) 
A (0) 2 3 41" 

4 



A (0 ) A (0 ) 
A (0) A 2 (0T- C ° S(Pi 2 5ine i " Pi 3 Sin6 + Pi 4 Sin9) 

J. u 



W A 3 (0 1 ) 

-nenrier cos((3i 2 sine - Pi 3 slne i + pi 4 sin9 - 2 V 

J- o 



VV A (6 )* 

a (0TA-T0T cos((3 V inG " Pi 3 sine + p V in V 

1 4 



W A 3 ,0 1 ) 

1~(e) A (0) cos(Pi 2 sin 0i - Pi 3 sin0 1 + Pi^in©) 



~K^Q) A 4 (0) cos (Pi ^^ - Pi 3 sin t Pi 4 sin ei f 2^) 



VV A 4 ' e i ) I 

^ 3 T0TA 4 l0T~' COS(,3i 2 Sin0 " P V ine ! + Pl 4 Sine i , | 



26 



[A (0 ) A (0 ) A (0 ) 
\m 1(8) v(e> —tfV-i " Pi 3 sine i " p V ine - V 



A (G ) A (9 ) A (B ) 
V(8) 1(8) A*<9) c ° s(! V in9 - P V ln9 l - P V in6 l ' V 



A (0 ) A (8 ) A (0 ) 

\ (9) 1(9) A* (9) c -«V ine i - ? V in6 * P V ln9 l " V 
12 4 



A (9 ) A (6 ) A (8 ) 

\ C 9 ) A (6) A*(9) <"»«V 1 '* 1 " P's'^! + "V 1 " 8 ! 
2 3 4 



,! 



, A (0 ) A (0 ) A (0 ) A (0 ) 
-4 I 1 1 2 1 3 1 4 1 ,n. . _ 

Tl) A o (0) A (9) A f8) " cos(Pi 2 Slne i 
12 3 4 



Pi sin0 



+ Pi sine, ) 

4 1 I 



(36) 



2 3 

There are no* 3 distinct interference patterns associated with S, S and S 

4 
in addition to the product pattern associated with S . It follows a fortiori 

that care must be exercised in the design of product patterns involving severa] 

multiplicative operations if interference is to be rejected. 

It is important to note that even in the case where time averaging is 

2 
possible, there are still terms in the S interference pattern that are not 

functions of £i t ) and hence cannot be averaged out. The terms of the S and 

S patterns are all functions of £.(t) and can be removed. If 6 is and 

the patterns ire normal ized, the time-averaged output reduces to 



iyi, o° s, e t v i 3 , i> 4 ) = i + s 2 [■ 1 ce 1 ) a. 2 (e 1 ) coscPi^mo^ 



B) coB(Pl 4 «in6 1 ) • :i 2 U) \ ) '•/ ,J i ) cos (Pi si no - Pi sine J 



27 



* (6,) MM cos (-(3| sine, + Pi .sine.) 
3141 31 41 



+ S 4 a ce n ) a (6. ) a O-.) a (0 ) cos (Pi sine - Pi sin0 * Pi sine. ) (37) 



in which we see that the S interference pattern is a sum of 4 cross products, 
e.g., a (0 ) a (0 ) cos(Pi sin - Pi sin ), of the original patterns 

^ J. *J3 1 ^ X *j X 

modified as usual by a cosine term. A useful design would minimize this 
pattern in the sidelobe region as well as the product pattern itself. 

We note also that if N is odd the random phase angles from the signal as 
well as the noise are present in all output terms and consequently time 
averages of the product pattern systems for N odd all go to zero. Hence, 
only an even number of antenna patterns can be cross-correlated in say a 
Radio Astronomy application. 



28 



4. SUPPRESSION OF THE EFFECT OF AVERAGE BACKGROUND NOISE BY 
USE OF THE NEGATIVE SIDELOBES OF THE PRODUCT PATTERN 



An interesting and useful feature of product patterns is their negative 
sidelobes. Thus they contrast with the power patterns of additive antennas 
which are non-negative functions of x = Pi sin 8./2„ It will be shown that 
for an arbitrary distribution of background noise in the absence of the signal 
the pattern can be adjusted so as to reduce to zero the time-averaged response 
to this noise in the system output. We assume that the noise coming from any 
direction is a narrow band quasi-stationary random process, statistically 
independent of the noise coming from any other direction. If its average 
power density is given by S (x, t) (which is generally a slowly varying unkno\ 
function), then the time-averaged response of the additive antenna system 
to this noise is 



I (x , t) = 
o' 



S 2 (x, t)|C (x - x)! 2 dx (38) 

7 o 



2 

where . C(x) " is the power pattern of the antenna with x corresponding to the 

o 

/ (3£ s i n 9 \ 2 
main lobe direction I x = — . S (x } t) is clearly non-negative and 

the integral will be positive for any noise background that is not identically 

zero for all x. 

For the produd pattern case, the time-averaged output is 



I (x , t) = 

J 



r 

S 2 (x, t) P o (x - x) dx (39) 






and since P, (x) can tak'.- on negative values it should be possible to select 

iny given time t the integral equation 



29 



r 

S 2 (x, t ) P (x - x) dx = (40) 

' o 2 o 



To make Equation (40; hold for all t it is clear that P (x) must become a 

slowly varying function of time in order to compensate for the variation 

2 

in S (x, t). Thus Equation (40) can be generalized to 

r 

S 2 (x, t) P (x - x, t) dx = (40a) 

J 2 o 

In addition, as the beam angle scans it will be necessary, in general, for the 
pattern to change if the output due to the noise is to remain at zero. 

In this derivation it is assumed that it is known a priori when the signal 
is present and when it is absent. During the periods when it is absent the 
antenna pattern is adjusted to eliminate the effect of the slowly varying 
average background noise. Then when the signal is transmitted its effect 
alone will be observed in the system's output, A slowly varying background 
noise density will gradually change this ideal condition and so there should 
be some prearranged sequence of time intervals in which the signal is turned 
off and the receiver pattern modified to compensate for this gradual change 
of background noise density. 

To illustrate, let us consider the Compound Interferometer with the uni- 
formly weighted aperture replaced by a linear array of m * 1 elements with 

real and symmetric weighting (a = a where r indicates the r element to 

r -r' 

the right from the center of the array and -r indicates the r element to 
the lef t; : The patterns of the two antennas are 

m/2 

A(x - x ) = 2_ a cos[2r(x - x )] (41) 

o r=0 r *• o 



Dl 



B(x - x ) = cos[ m(x - x )] 
o o 



30 



(42) 



R« ■ n Pi sin 

Pi sin 9 o 

where x = - — — ■ — — , x = ■ 

2 ' o 2 



and G is the beam angle of the patterns, 
o 



Now from Equation (3a) it follows that 



m/2 



P (x - x ) = 2 a cosi r(x - x )] cos[m(x - x )] cos[2m(x - x )] 
2 o r o o o 

r=0 



(43) 



m/2 



m/2 



= 2af(x-x)=2 af(x-x) 
rr o rro 

r=0 r=0 



Substituting this into the expression for the output due to the arbitrary 
background noise density (Equation (39)) we get 



I(x , t) = 

o' 



m 2 
= 2 
r=0 



2 m/2 
S (x, t) 2 a f (x - x) d: 
rro 
r=0 



S (x, t) f (x - x) dx 
r o 



(44) 



and from Equation i40) we require that 



m 2 
2 a 
, ' 



2 m/2 

S x t) f fx - x) dx = 2 a q (x , t) = 
r o r r o' 

r=0 



(45) 



where q (x t ) 
r o 



S (x, t) f (x - x ) dx 
r o 



(46) 



To get the coil I , q (x , l) it is necessary to install switches in the 

r o 

trar. from the various array elements. If all switches but the 

the center element' I Lne were open then the output from the system 

i 



31 



I (x , t) = a q (x , t) (47) 

o o o o o 



and if during the measurement starting at, say, t , a is set equal to unity, 
we get 

q (x t ) = I (x t ) (47a) 

o <r o o o o 



Likewise opening; all but the lines from the a elements and setting a equal 
ft. r r 

to unity we get 



q (x , t ) = I (x , t ) (48) 

r o' o r o o 



In this way all of the coefficients can be obtained. 

We can think of the q (x , t)s as components of an — + 1 dimensional noise 

r o- 2 

vector Q'x t 1 whose scalar product with the weighting vector A('x , t ) also 
o o o 

f _ a. i dimensions, is required to be zero by Equation (45) „ It is clear 

that there is no unique vector A(x , t ) that will satisfy this condition since 

o o 

any vector will do which lies in the hyperplane to which Q<x t ) is perpen- 

o o 

dicular Out of this infinity of vectors that satisfy Equation (45) it would 

be best to select one which gave the maximum response to the signal when it 

was incident at the beam angle 6 . This means that we pick Aix t ) such that 

o o o 

m/2 
P (0) = 2 a (x , t ) = MAX (49) 

2 r=0 



However it is obvious that some limit on the size of the vectors A('x , t ) 

o o 

themselves must be established because if any vector is doubled in size then 
the sum of its components will also be doubled, Hence we seek out from all 



32 

vectors of a given length lying in the hyperplane the one whose sum of its 
components is the largest. We set this length (without loss of generality) 
equal to unity and we obtain the condition equation 

m/2 

2 a (x , t ) = 1 (50) 

r=0 r ° ° 

Note that there are — + 1 coefficients a (x , t ) and still only 3 

2 r o o 

simultaneous equations. There is thus a possible — - 2 additional conditions 
that could be satisfied, e.g. 1) first null occurs at 9 - A, 2) first 
sidelobe has maximum at = TT , etc. 

However, let us consider the five-element case, m = 4 in which there 
are just three possible conditions; the three equations are 

a (x , t ) + a, (x , t ) 4 a (x , t ) = MAX 

o o' o loo 2 o 7 o 

c (x , t ) a fx , t ) + c, (x , t ) a (x , t ) f c (x , t ) a <x , t ) - 
oooooo lo'oloo 2 o o 2 o o 

2 2 2 

a (x , t ) + a, (x , t)+a„(x.t)=l 
o o' o loo 2o'o 

(51) 

The last two equations represent a plane and a sphere, respectively, in three 
dimensional space. Their intersection is a circle of unit radius centered at 
the origin and a certain point (or points) on it satisfy the first equation. 
To get this point I arrange' s method of undetermined multipliers can be used 
and the following s< I -,\ equal ions results. 

L + c (x , t ) X (x , t ) + 2a (x , t ) X (x , t ) » 

-/ o o o O o o 



33 



1 .. c ix , t i X (x , t ) + 2a, (x . t ) \ (x t ) = 
loo ooo loo lo o 

(52) 

1 - c (x , t ) v 'X , t ) + 2a (x , t ) X (x , t ) = 
2 " o o ooo 2 o o loo 

c (x , t ) a <x , t ) + c, (x , t ) a n (x , t ) + c„ (x t ) a (x t ) = 

ooo o o o 1 o' o 1 o' o 2 o 7 o 2 o o 

2 2 2 

a ix t ) + a (x , t ) + a„ (x , t ) = 1 

o o' o 1 o' o 2 o' o 

A digital computer could be used to solve these equations for the weighting 

coefficients a ( x s t ) a (x , t) and a (x , t ) Then instructions from 
ooolo'o 2 o o 

the computer could be fed back to each of the lines coming from the array 

elements where a variable attenuator would be adjusted to give the correct 

weighting. A diagram of the system is shown in Figure 5. Note that the 

computer also sends phase shift instructions (for scanning) and switching 

instructions i for obtaining the q (x , t ) coefficients) The conditions 

r o o 

fed into the computer are, of course, those that are mentioned above and 
which give rise to Equations (52) which the computer must solve. 

A final word of caution should be mentioned Although in theory the 
equations can always be solved to reduce the noise output to zero, some noise 
backgrounds can be eliminated only by resorting to a supergain condition For 
example, a not uncommon case is that of a single point source of noise which 
can be eliminated only if the pattern always has a null in the direction of 
the point source This presents no difficulty until the beam angle x itself 
approaches the angle of the point source. This requires the null to approach 
the beam angle, and since the pattern is an analytic function the response 
of the pattern to the signal incident at the beam angle will approach zero. 




V 

3 

r-H 
> 

CO 
bfl 
cd 

0) 

> 

< 

CO 

x: 
+j 

<H 

O 

c 
o 

•H 
CO 
W 



a 
<x 

3 
CO 

(1) 
XI 



0J 

in 

•H 

o 

c 

•o 

c 

3 

o 

bD 

o 

CQ 
<H 

o 

c 
o 

•H 

+J 
3 
X3 



cd (-< 

O -H 



U 

o 

a) o 
-t-> 

en co 

>> A 

CO -P 



(D 

3 
be 

•H 



3 >> 

a m 
•m cd 

3£ 



35 

To maintain the response at, say, unit amplitude it will be necessary to 
increase the magnitude of the weighting coefficients a (x , t ) which means 
going to a supergain condition. Furthermore, if the beam angle and the angle 
of the point source of noise coincide it will be necessary to split the beam to 
satisfy Equation (45). This means that it is impossible to eliminate the noise 
and not the signal if both are incident as plane waves at the same angle. How- 
ever, there is usually more than one source of noise and, in theory at least, 
the signal from a point source which may or may not coincide with one of the 
noise sources can be retained and the noise rejected. 



36 



5. THE NOISE LIMITATION ON THE FORMATION OF ARBITRARY PATTERNS 
FROM A TWO ELEMENT INTERFEROMETER BY 
MEANS OF MULTIPLICATIVE OPERATIONS 

5,6 
Several writers •' have discussed the possibility of forming arbitrary 

antenna patterns by multiplicative operations on the cross-correlated output 

from a single pair of elements spaced a distance I apart^ i.e., from a simple 

interferometer „ Patterns of arbitrary sharpness of the main lobe could be 

synthesized and used to advantage in direction-finding applications. However^ 

no account was taken of noise in these schemes^ and it will be shown below 

that noise severely limits their usefulness. 

From Equation (3a) the pattern of the interferometer is 



P (6, i) = cos (Pi sin 8) 



(58) 



It is raised to various powers from say to M and a linear combination of 
these terms each weighted by a factor q is formed 



M k 

Q (x) = 2 q cos x 
k=0 



(59) 



where x = Pi sin 9 But 



k 
cos X 



k-1 




k 



1 l k 
cos[ (k - 2r)x] -4 — 

„ k k 

2 la 



(60) 



if k is even, and 



k 
COS X 



k-1 

2 | ') 



r, 



cos[ (k - 2r) x] 



(60a) 



37 



if k is odd and where 



is the binomial coefficient. Hence 



M 
Q (x) = 2 w cos(kx) 
k=0 



where the w s are linear combinations of the q.'s. By a proper choice of the 
k K 

q :'s any arbitrary pattern of a 2M + 1 element array (M even) can be simu- 
lated. If M is odd the array has 2M elements. 

The assumption is made that there is no noise. In particular, it is 
assumed that only a single plane wave ; the signal ; is incident on the antennas, 
In the following it will be shown that if the signal is received in an 
arbitrary background of noise there will generally be a shift in the apparent 
direction of the signal as "seen" by the basic pattern cos (Pi sin 0) and the 
process of forming a "better" pattern by multiplicative operations cannot 
remove this error. 

Let the signal of say unit average power be incident on the system from 

direction x = Pi sin 9 and let the noise have an average power density given 

2 

by S (x) . Then the time-averaged output from the low-pass filter of the 

correlation system is 



I(x) = 

o 



;5( X;l - x) + S^( Xl ), 



cos(x - x ) dx 
oil 



cos(x - x) + 
o 



S (x, ) cos x dx 
1 11 



cos X 



S (x ) sin x dx 



sin x 



(61) 



where x is the beam angle of the interferometer pattern. We see that the 



38 



terms in the square brackets are the Fourier coef f icients, a and b , of the 
series expansion of S(x) given by 



where 



S(x) = Z a cos kx + 2 b sin k x 

k=o k k=i k 



cos(kx) cos(jx) dx = 6 



sin(kx) sin(jx) dx = 6, , 

kj 



sin(kx) cos(jx) dx = , 



and 



5 kj =l, k-J 



= 0, k £ j 



Therefore Equation (60) can be expanded as 



I < x ) = cos(x - x) + a cos x + b sin x 
o o 1 o i o 



= cosCx - x) + c cos(x - x ) 
o l o c 



where 



2 2 . 2 . -1 
c, = a, + b, , x = tan 

1 1 1 c 



The arbitrary nod ie background is "seen" by the pattern as a point source at 



39 



ang 



le x = Pi sin with average power c . 
c c 1 

Equation (63) can be reduced finally to 



I(x ) = [ cos x + c cos x 1 cos x + [ sin x + c sin x ■] sin x 
o 1 c J o 1 c o 



d, cos(x - x ) 
1 o d 



(66) 



where 



2 ,2 

[cos x + c cos x + [sin x + c, sin x J 
L l c J L 1 c J 



x = tan 
d 



-1 



sin x + c sin x 
1 c_ 

cox x + c cos X 
1 c. 



(67) 



The relations are best shown by the phasor diagram of Figure 6. The output is 

jx 
the scalar product of the unit scanning phasor e and the phasor sum e ' + 

J x ^ J x h 
c d 

c e = de There is a magnitude error and an angular error in the esti- 
mated value of the signal. However^ if x = x or x -77 the angular error is 



ze 



ro; the magnitude error is maximized and is - c. . Contrariwise if x = 

1 c 

x - cos (c ,/2) the amplitude of d is unity and no magnitude error occurs. 

The maximum error in angle for a given c < 1 is (x -x) =- sin c 

1 d max 1 

and occurs at x = x + (77/2 + sin e, ) . If c ' • 1 the error can be as much 
c 11 — 

as 77, 



The basic patterns due to the signal alone ; the noise alone and the signal 
plus noise are shown in Figures 7a, 7b ? and 7c respectively. In Figure 7d a 
pattern Q„(x - x ) synthesized from the basic signal plus noise pattern. 

MOd or- r , 

d cos(x - x ) is shown. The error (x, - x ) occurring in the basic pattern 
1 o d d 

can in no way be reduced by this synthesizing of Q (x - x ) since the process 

Mod 



40 




0) 






a 


SI. 


in o 


o 


c 


3 Z 


M 





H 


^< 


—i 


0. in 


o 


< 


3 


x-a 




— 1 r-l 


(U 


CO CU 


c 


05 


a 


bO 


•^1 


60 r-l 


i-( 





■H ffl 


CO 


z 


CO C 

bo 


<M 


VI 


<H -H 








O CO 


c 


c 


C <H 


u 


u 


U O 


0) 


4) 


0) 


4-> 


•P 


P c 


■p 


■»-> 


P u 


ed 


0) 


3) CO 


0, 


£ 


a, *> 
p 


k >-• 


t< 


in at 


1 0) 


0) 


a) a, 


•p 


■u 


p 


0) 


0) 


d) T3 


s 


G 


E a» 


■s. o 





« 


V ^ 


l* 


U -H 


-^ 0> 


CO 


0) in 


<H 


<*-{ 


<H 0) 


u 


^ 


fc £ 


<D 


0) 


1) P 


+) 


P 


P c 


—^ a 


G 


c >> 


J w 


i—i 


HH CO 



d A u Q 



D 



O 



"O 



















\ 






















\\ 










tr 












M, 








UJ 
CO 




CO 

< 


1 
1 


















z 


I 


w 1 

X *J 
^1 










a: 


\ 








0" 


/ / 






^ 


_l 

z 

CO 
CO 


O 

CO 

< 

X 
0. 


\ 

1 


\ 


\ \ 






AV 














\ 


\ \ 






/ \ 














\ 


^ 




' / 




































\ 






^/ 























^ 


^ — 










— \ 


X 


\ 
\ 


z / 






\ 






"O 






\ 2 


Z CO / 








V 










*— *• 


\ 


< <\/ 










\ 








X\ 




1 7\. 












\ 






1 


\ X ^«- — ► 


to Q- / ' 














^ 


Xy 


w\S\ 


















\ 


\ 


\ \ "°v 

























•-* 


G 




ct) 


V 




C 


B 




hfi 


0) 




-H 


r-H 




w 


W 




CO 


O 




JG 

p 


^ 




<H 


m 




O 


a 




c 










h 


>. 


■H 


m 


c« 


*-> 




^ 


fit 


*-> 


U 


P 


D 


< 


C 


D. 




CO 


♦^ 


<x> 




3 


> 


t< 




*-> 


a 


<u 


Cfl 


<D 


Ul 


U 


IX 


•H 


H 




O 


r-< 


u 


^. 


D. 







•H 


'/i 


U5 


•(-> 


Tj 


g 


rH 


x: 


.-1 


3 


E 


a 


fl 


'D 






Cb 






^ 






3, 







41 



only defines more sharply the value of x for the benefit of the observer. 
The synthesized pattern is limited not by its own shape and sharpness (in 
theory it can be made to approach a delta) but by the basic interferometer 
pattern to which any combination of point source plus background noise 
appears as another equivalent point source. This is all that the synthesized 
pattern will ever "'see". If there is a fairly low signal to noise ratio it 
can be concluded that a method of pattern syntheses such as this can lead 
to large errors in the estimated value of the angle of incidence of the 
signal . 



42 
6. CONCLUSIONS 

It has been shown that the cross-correlation of antenna voltages to 
obtain product patterns gives rise to two major problems: 

1. Multiplicative distortion of amplitude modulation, 

2. Interference patterns which differ from the original product pattern 
and which can have high sidelobes even when the product pattern itself has 
low sidelobes. 

The first problem can in theory be overcome by a rather complicated 
system which requires frequency shifting and filtering. An analysis of the 
second problem shows that if instantaneous reception of, say, modulated signals 
is required, special care must be exercised in the design of both product 
and interference patterns if an adequate rejection of interference is to be 
achieved. For systems in which the time average of the incoming signals is 
all that is required the problem simplifies considerably but if N, the number 
of patterns to be multiplied, is large it is still quite involved. 

It has also been shown that product patterns possess a distinct advantage 
over conventional additive power patterns in that their negative sidelobes can 
be used to suppress the integrated effect of any arbitrary distribution of 
background noise. 

Finally, the artificial formation of arbitrarily sharp antenna patterns 
by multiplicative operations on the time-averaged output of a two element 
interferometer system has been shown to be of limited value in the presence 
of ,i background noise distribution because the noise causes a shift in the 
apparent direction of arrival of the signal. This shift is not reduced by 

formation '<( i more directive" pattern whose effect is therefore nothing 



43 

more than to emphasize the apparent direction of arrival of the signal as 
indicated by the interferometer pattern. If the noise is relatively strong 
the error in the apparent direction of arrival can be quite large. 



44 
REFERENCES 



1. Mills, B. Y. and Little, A. G., "A High-Resolution Aerial System of a 
New Type." Austral. J. Phys. , Vol. 6, pp. 272-278, September, 1953. 

2. Covington, A. E. and Broten, N. W "An Interferometer for Radio 
Astronomy with a Single Lobed Radiation Pattern," IRE Trans, PGAP, 
AP-5, No. 3, pp. 247-255, July, 1957. 

3. Welsby, V, G. and Tucker, D. G. , "Multiplicative Receiving Arrays," 
J, Brit, IRE, 19, pp. 369-382, June, 1959 

4. Berman, A. and Clay, C. S., "Theory of Time-Averaged Product Arrays," 
J. Acoustical Soc. America , 29, No. 7, pp. 805-812, August, 1957. 

5. Drane, C. J,, Jr., "Phase Modulated Antennas," presented at IRE-URSI 
Joint Fall Meeting, 1959, San Diego, California. 

6. Brown, J. L M Jr. and Rowlands, R. 0., "Design of Directional Arrays,' 
J. Acoustical Soc. America , 31, No. 12, pp. 1638-1643, December, 1959. 

7. Dwight, H. B., Tables of Integrals and Other Mathematical Data , 
The Macmillan Company, p.l. 



ANTENNA LABORATORY 
TECHNICAL REPORTS AND MEMORANDA ISSUED 



"Synthesis of Aperture Antennas, " Technical Report No : 1, C.T.A, Johnk, 
October. 1954. * 

A Synthesis Method for Broad-band Antenna Impedance Matching Networks," 
Technical Report No. 2, Nicholas Yaru, 1 February 1955 

The Asymmetrically Excited Spherical Antenna," Technical Report No, 3, 
Robert C Hansen. 30 April 1955,* 

"Analysis of an Airborne Homing System," T ech nical Report No, 4, Paul E„ 

Mayes r 1 June 1955 ( CONFIDENTIAL) . 

Coupling of Antenna Elements to a Circular Surface Waveguide, Technical 
Report No 5, H. E King and R. H. DuHamel , 30 June 1955.* 

Axial ly Excited Surface Wave Antennas," Technical Report No. 7, D, E. Royal, 
10 October 1955 * 

Homing Antennas for the F-86F Aircraft f 450-2500mc) , " Technical Report No. 8, 
P.E t Mayes R.F, Hyneman, and R.C. Becker, 20 February 1957, (CONFIDENTIAL). 

Ground Screen Pattern Range," Technical Memorandum No 1, Roger R. Trapp, 
10 July 1955 * ~~~ ~~~ 



Conj rag t AF33 ( 61 6 i ^3220 

Effective Permeability of Spheroidal Shells," Technical Report No. 9, E. J. 
Scott and R H DuHamel, 16 April 1956. 

An Analytical Study of Spaced Loop ADF Antenna Systems," Technical Report 
No. 10, D G Berry and J B, Kreer, 10 May 1956. 

"A Technique for Controlling the Radiation from Dielectric Rod Waveguides," 
Tech_nical_Repot No. 11, J W. Duncan and R = H DuHamel, 15 July 1956.* 

Directional Characteristics of a U-Shaped Slot Antenna, ' Technical Report 
No^_12, Richard C Becker, 30 September 1956 „** 

Impedance of Ferrite Loop Antennas," Technical Report No 13, V, H, Rumsey 
and W. L Weeks. 15 October 1956 

"Closely Spaced Transverse Slots in Rectangular Waveguide," Technical Report 
No^_l_4, Richard F Hyneman, 20 December 1956. 



Distributed Coupling to Surface Wave Antennas," Technical Report No 15. 

jj^ ' — — — * 

Ralph Richard Hodges, Jr , 5 January 1957. 

The Characteristic Impedance of the Fin Antenna of Infinite Length, Technical 
Report No 16 Robert 1 Carrel, 15 January 1957 

On the Estimation of Ferrite Loop Antenna Impedance, Technical Re port No. 17, 
Walter L. Weeks, 10 April 1957.* 

A. Note Concerning a Mechanical Scanning System for a Flush Mounted Line Source 
Antenna fechnic .1 Report No. 18, Walter L. Weeks, 20 April 1957, 

Broadband logarithmically Periodic Antenna Structures. Technical Report No . 19 
R K DuHamel and D E. Isbell, 1 May 1957. 

Frequency Independent Antennas," Tec hnical Report No 20 V. H Rumsey, 25 

October 1957 

The Equiangular Spiral Antenna," Techni cal Report No, 21, J D, Dyson, 15 
September 195? 

Experimental Investigation of the Conical Spiral Antenna Technical Report 
Vo 22. R L. Carrel 25 May 1957.** ' """ 

Coupling between a Parallel Plate Waveguide and a Surface Waveguide," Technical 
Report \ T o. 23 E ' Scott, 10 August 1957. 

Launching Efficiency of Wires and Slots for a Dielectric Rod Waveguide," 
rechni .. | i 24 J. W. Duncan and R. H DuHamel, August 1957. 

The Characteristic Impedance of an Infinite Biconical Antenna of Arbitrary 
Cross Section, hnical Repo rt No. 25, Robert L Carrel, August 1957. 

-Backed Slot Antennas, ' Technical R eport No. 26_, R. J. Tector, 30 
October 1957 

Coupled Waveguide Excitation of Traveling Wave Slot Antennas/ Techni cal 
Re] [0 27 W I Weeks, 1 December 1957. 

Pti Red ingular Waveguide Partially Filled with Dielectric," 

28 W I Weeks, 20 December 1957 

p< r lint Length of Biconical Structures of Arbitrary 
Cross rechnli i] Rep ort No. 29 , J. D Dyson. 10 January 1958 

Lcally Periodic Antenna Structur Technical Report No. 30, 
D ^58 

( ' pular Slots,' technical Reporl ^'< 31, N. J. 
; , L958 

tation oi i Surface Wave on a Dielectric Cylinder/' 
I w Du J5 May L058 



"A Unidirectional Equiangular Spiral Antenna, " Technical Report No. 33, J. D. 
Dyson, 10 July 1958 

Dielectric Coated Spheroidal Radiators," Technical Report No. 34^ W. L Weeks, 
12 September 1958. 

"A Theoretical Study of the Equiangular Spiral Antenna," Technical Report 
No. 35, P E Mast, 12 September 1958. 



Contract AF33 '616) -6079 

Use of Coupled Waveguides in a Traveling Wave Scanning Antenna,' Technical 
ILERELLJ^JL 6 -.' R H MacPhie, 30 April 1959. 

On the Solution of a Class of Wiener-Hopf Integral Equations in Finite and 
Infinite Ranges, Technical Report No. 37, Raj Mittra, 15 May 1959. 

"Prolate Spheroidal Wave Functions for Electromagnetic Theory," Technical 
Repojr£_No ^_38 , W, L, Weeks, 5 June 1959. 

Log Periodic Dipole Arrays," Technical Report No. 39, D E Isbell, 1 June 1959. 

A Study of the Coma-Corrected Zoned Mirror by Diffraction Theory," Technical 
Report No^ 40. 3. Dasgupta and Y. T„ Lo, 17 July 1959 

The Radiation Pattern of a Dipole on a Finite Dielectric Sheet/" T echnical 
ReportNq 41 KG Balmain, 1 August 1959 

The Finite Range Wiener-Hopf Integral Equation and a Boundary Value Problem 
in a Waveguide. Technical Report No. 42, Raj Mittra, 1 October 1959. 

"impedance Properties of Complementary Mul titerminal Planar Structures " 
Te chjrica.1 JRepor t No __43_, G, A, Deschamps, 11 November 1959 

On the Synthesis of Strip Sources," Technical Report No 44 Raj Mittra, 
4 December 1959 

Numerical Analysis of the Eigenvalue Problem of Waves in Cylindrical Waveguides," 
Technical Report No^ 45. C H Tang and Y. T, Lo, 11 March I960, 

Ve* Circularly Polarized Frequency Independent Antennas With Conical Beam or 
Omnidirectional Patterns," Tec hnica l Report No, 46, J.D Dyson and P.E. Mayes, 
20 June 1960 

Logarithmically Periodic Resonant-V Arrays,'" Technical Report No, 47, P.E. 
Waves and R L Carrel, 15 July 1960. 

* Copies available for a three week loan period 
Copies no longer available. 



AF 33(616) -6079 



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Attn; W, A, Kee, Chief Librarian 

M/F Contract AF33(600)-37705 
Library & Document Section 
Baltimore 3, Maryland 

Ennis Kuhlman 

McDonnell Aircraft 

P.O. Box 516 

Lambert Municipal Airport 

St, Louis 21, Missouri 

Melpar, Inc . 

Attn, Technical Library 

M/F Contract AF19(604)-4988 
Antenna Laboratory 
3000 Arlington Blvd. 
Falls Church, Virginia 

Melville Laboratories 
Walt Whitman Road 
Melville, Long Island, 
New York 



Motorola, Inc , 

Attn: R, C, Huntington 
8201 E, McDowell Road 
Phoenix, Arizona 

Physical Science Lab. 

Attn. R. Dressel 
New Mexico College of A and MA 
State College, New Mexico 

North American Aviation, Inc , 

Attn: J. D Leonard, Eng . Dept . 
M/F Contract NOa(s) 54-323 
4300 E, Fifth Avenue 
Columbus, Ohio 

North American Aviation, Inc, 
Attn: H, A. Storms 

M/F Contract AF33( 600)- 36599 
Department 56 
International Airport 
Los Angeles 45, California 

Northrop Aircraft, Inc. 

Attn: Northrop Library, Dept. 2135 
M F Contract AF33 (600)-27679 
Hawthorne, California 

Dr, R. E, Beam 
Microwave Laboratory 
Northwestern University 
Evanston, Illinois 



AF 33' 616'. -6079 



Ohio State University Research 
Foundation 

Attn: Dr, T, C, Tice 

M F Contract AF33(616)-6211 
1314 Kinnear Road 
Columbus 8, Ohio 

University of Oklahoma Res, Inst. 
Attn Prof, C, L, Farrar 

M F Contract AF33( 616)-5490 
Norman, Oklahoma 



Dr. D. E. Royal 

Ramo-Wooldridge, a division of Thompson 

Ramo Wooldridge Inc, 

8433 Fallbrook Avenue 

Canoga Park, California 

Rand Corporation 
Attn: Librarian 

M/F Contract AF18(600)-1600 
1700 Main Street 
Santa Monica, California 



Philco Corporation 

Government and Industrial Division 
Attn Dr, Koehler 

M. F Contract AF33(616)-5325 
4700 Wissachickon Avenue 
Philadelphia 44, Pennsylvania 

Prof, A A, Oliner 
Microwave Research Institute 
Polytechnic Institute of Brooklyn 
55 Johnson Street - Third Floor 
Brooklyn, New York 

Radiation, Inc , 
Technical Library Section 
Attn Antenna Department 

M F Contract AF33(600)-36705 
Melbourne, Florida 

Radio Corporation of America 
RCA Laboratories Division 
Attn Librarian 

M F Contract AF33( 616)-3920 
Princeton, New Jersey 

Radioplane Company 

M/F Contract AF33( 600) -23893 

Van Nuys, California 

Ramo-Wooldridge, a division of 
Thompson Ramo Wooldridge, Inc. 

Attn technical Information Services 
8433 Fallbrook Avenue 
P 0. Box 1006 
Canoga Park California 



Rantec Corporation 

Attn: R. Krausz 
M/F Contract AF19( 604)-3467 
Calabasas, California 

Raytheon Manufacturing Corp „ 
Attn; Dr, R, Borts 

M/F Contract AF33(604)-15634 
Wayland, Massachusetts 

Republic Aviation Corporation 
Attn: Engineering Library 

M/F Contract AF33( 600)-34752 
Farmingdale 
Long Island, New York 

Republic Aviation Corporation 
Guided Missiles Division 
Attn: J. Shea 

M/F Contract AF33(616)-5925 
223 Jericho Turnpike 
Mineola, Long Island, New York 

Sanders Associates, Inc. 
95 Canal Street 

Attn: Technical Library 
Nashua, New Hampshire 

Smyth Research Associates 

Attn.; J. B. Smyth 
3555 Aero Court 
San Diego 11, California 

Space Technology Labs, Inc, 

Attn; Dr., R. C Hansen 
P.O. Box 95001 
Los Angeles 45, California 
M/F Contract AF04( 647 ) -361 



AF 33(616^-6079 



Sperrv Gyroscope Company 
Attn B, Berkowitz 

M. F Contract AF33( 600)-28107 
Great Neck 
Long Island, New York 

Stanford Electronics Laboratory 
Attn. Applied Electronics Lab. 
Document Library 
Stanford Iniversity 
Stanford, California 

Stanford Research Institute 

Attn; Mary Lou Fields, Acquisitions 
Documents Center 
Menlo Park, California 

Stanford Research Institute 
Aircraft Radiation Systems Lab, 
Attn D, Scheuch 

M F Contract AF33(616)-5584 
Menlo Park, California 

Sylvania Electric Products, Inc. 
Electronic Defense Laboratory 
M/F Contract DA 36-039-SC-75012 
P.0, Box 205 
Mountain View, California 

Mr, Roger Battie 

Supervisor, Technical Liaison 

Sylvania Electric Products, Inc, 

Electronic Systems Division 

P.O. Box 188 

Mountain View, California 

Sylvania Electric Products, Inc , 
Electric Systems Division 
Attn- C. Faflick 

M/F Contract AF33(038)-21250 
100 First Street 
Waltham 54, Massachusetts 



Technical Research Group 
M/F Contract AF33< 61 6 ) -6093 
2 Aerial Way 
Syosset, New York 

Temco Aircraft Corporation 
Attn: G, Cramer 

M/F Contract AF33( 600) -36145 
Garland, Texas 

Electrical Engineering Res, Lab, 
University of Texas 
Box 8026, University Station 
Austin, Texas 

A, S 5 Thomas, Inc„ 
M/F Contract AF04C 645;-30 
161 Devonshire Street 
Boston 10, Massachusetts 

Westinghouse Electric Corporation 
Air Arm Division 
Attn: P, D, Newhcuser 

Development Engineering 
M/F Contract AF33(600)-27852 
Friendship Airport 
Baltimore, Maryland 

Professor Morris Kline 

Institute of Mathematical Sciences 

New York University 

25 Waverly Place 

New York 3, New York 

Dr , S. Dasgupta 

Government Engineering College 

Jabalpur, M.P, 

India 

Dr , Richard C. Becker 
10829 Berkshire 
Westchester, Illinois 



Tamar Electronics, Inc, 
Attn LB McMurren 
2045 W Rosecrans Avenue 
Gardena, California 



The Engineering Library 
Princeton University 
Princeton, New Jersey 



AF 33(616)-6079 

Dr. B, Chatterjee 
Communication Engineering Dept , 
Indian Institute of Technology 
Kharagpur (S.E. Rly,) 
India 

Sperry Phoenix Company 
Attn: Technical Librarian 
P.O. Box 2529 
21111 North 19th Avenue 
Phoenix, Arizonia 

Dr. Harry Letaw, Jr„ 

Raytheon Company 

Surface Radar and Navigation Operations 

State Road West 

Wayland, Massachusetts